Analytic Derivation of the Gravitational Constant (G): Dimensional Dilution and Tensor Volumetric Distortion in a 6D Topological Manifold

Contemporary theoretical physics continues to explore the fundamental analytic challenge regarding the existence of free parameters within the Standard Model, which are generally determined through empirical measurement rather than derived directly from fundamental mathematical principles. As a complementary effort to augment this phenomenological framework, this manuscript proposes the Connecting Thread Theory (TBP), a deductive approach based on a deterministic geometric spatial layout (zero-free-parameter framework). This study investigates the framework wherein the fundamental constants of the universe are not independent entities, but rather can be derived analytically from the topological structure of a 6-Dimensional manifold (d=6). Through the mapping of an axiomatic constitution, this model attempts to formulate the values of physical constants as logical consequences of spatial resolution and mechanical interactions of the vacuum. Based on the analytic derivations within this matrix system, several fundamental parameters are evaluated with exact fractional precision, including: Active Circuit Load: \mu = 88 Vacuum Energy Ceiling (Micro Blackhole / VEV limit): R_{\text{crit}} = \frac{1936}{9} \text{ GeV} = 215.111111111... \text{ GeV} Observational Electromagnetic Density (Inverse Fine-Structure Constant): \alpha^{-1}_{\text{obs}} = 137.036000... Strong Force Density (Strong Coupling): \alpha_s = \frac{17}{144} = 0.1180555... Geometric Friction Limit: H_{\text{lim}} = \frac{1}{528} This manuscript presents the deductive construction of the aforementioned parameters, ranging from the foundational geometric genealogy to the calculation of matrix backreaction. This geometric constitution approach aims to expand upon established phenomenological models by offering an analytically traceable algebraic genealogy for the parameters of force and matter.